scholarly journals GENERALIZED JACOBI REPRODUCING KERNEL METHOD IN HILBERT SPACES FOR SOLVING THE BLACK-SCHOLES OPTION PRICING PROBLEM ARISING IN FINANCIAL MODELLING

2018 ◽  
Vol 23 (4) ◽  
pp. 538-553
Author(s):  
Mohammadreza Foroutan ◽  
Ali Ebadian ◽  
Hadi Rahmani Fazli
Keyword(s):  
Option Pricing ◽  
Reproducing Kernel ◽  
Initial Conditions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Kernel Functions ◽  
Dirichlet Boundary Conditions ◽  
Financial Modelling ◽  
Black Scholes ◽  
Reproducing Kernel Theory

Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

scholarly journals A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kdv Equation ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
The Kdv Equation ◽  
Novel Method ◽  
Reproducing Kernel Theory

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

scholarly journals Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

2020 ◽  
Vol 4 (2) ◽  
pp. 27 ◽  
Author(s):  
Onur Saldır ◽  
Mehmet Giyas Sakar ◽  
Fevzi Erdogan
Keyword(s):  
Fractional Order ◽  
Reproducing Kernel ◽  
Burgers Equation ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Neumann Boundary ◽  
Kernel Functions ◽  
Iterative Approach ◽  
Reproducing Kernel Method ◽  
Reproducing Kernel Spaces

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

scholarly journals Numerical Simulation Characteristics of Logging Response in Water Injection Well by Reproducing Kernel Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Ming-Jing Du ◽  
Yu-Lan Wang ◽  
Temuer Chaolu
Keyword(s):  
Temperature Field ◽  
Water Temperature ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Water Injection ◽  
Kernel Functions ◽  
Injection Well ◽  
Two Phase Flows ◽  
Two Phase

Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed byn-term summation. The satisfied numerical results were carried out by Mathematica 7.0, showing that the larger the difference between injected water temperature and initial borehole temperature or water injection conditions, the more obvious the indication of water accepting zones. The numerical examples evidence the feasibility and effectiveness of the proposed method of the two phase flows in engineering.

scholarly journals Explicit Solution of Telegraph Equation Based on Reproducing Kernel Method

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Exact Solution ◽  
Explicit Solution ◽  
Reproducing Kernel ◽  
Initial Conditions ◽  
Kernel Method ◽  
Telegraph Equation ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Reproducing Kernel Theory ◽  
Kernel Theory

We propose a reproducing kernel method for solving the telegraph equation with initial conditions based on the reproducing kernel theory. The exact solution is represented in the form of series, and some numerical examples have been studied in order to demonstrate the validity and applicability of the technique. The method shows that the implement seems easy and produces accurate results.

scholarly journals Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shatha Hasan ◽  
Nadir Djeddi ◽  
Mohammed Al-Smadi ◽  
Shrideh Al-Omari ◽  
Shaher Momani ◽  
...  
Keyword(s):  
Reproducing Kernel ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Real Life ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Point Of View ◽  
The Novel ◽  
Life Problems ◽  
Fractional Models

AbstractThis paper deals with the generalized Bagley–Torvik equation based on the concept of the Caputo–Fabrizio fractional derivative using a modified reproducing kernel Hilbert space treatment. The generalized Bagley–Torvik equation is studied along with initial and boundary conditions to investigate numerical solution in the Caputo–Fabrizio sense. Regarding the generalized Bagley–Torvik equation with initial conditions, in order to have a better approach and lower cost, we reformulate the issue as a system of fractional differential equations while preserving the second type of these equations. Reproducing kernel functions are established to construct an orthogonal system used to formulate the analytical and approximate solutions of both equations in the appropriate Hilbert spaces. The feasibility of the proposed method and the effect of the novel derivative with the nonsingular kernel were verified by listing and treating several numerical examples with the required accuracy and speed. From a numerical point of view, the results obtained indicate the accuracy, efficiency, and reliability of the proposed method in solving various real life problems.

scholarly journals Numerical Solutions of the Second-Order One-Dimensional Telegraph Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kılıçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Telegraph Equation ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Reproducing Kernel Theory ◽  
Space Method

We investigate the effectiveness of reproducing kernel method (RKM) in solving partial differential equations. We propose a reproducing kernel method for solving the telegraph equation with initial and boundary conditions based on reproducing kernel theory. Its exact solution is represented in the form of a series in reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of this method. The results obtained from this method are compared with the exact solutions and other methods. Results of numerical examples show that this method is simple, effective, and easy to use.

scholarly journals Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator

2021 ◽  
Author(s):  
omar abu arqub ◽  
Jagdev Singh ◽  
Banan Maayah ◽  
Mohammed Alhodaly
Keyword(s):  
Reproducing Kernel ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Reproducing Kernel Hilbert Space ◽  
Research Work ◽  
Three Dimensional ◽  
Characterization Theorem ◽  
Space Technique ◽  
Kernel Approach ◽  
Reproducing Kernel Theory

In this research study, fuzzy fractional differential equations in presence of the Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing Kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions; analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

scholarly journals Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation

2018 ◽  
Vol 8 (2) ◽  
pp. 145-151 ◽  
Author(s):  
Esra Karatas Akgül
Keyword(s):  
Hilbert Space ◽  
Inverse Problem ◽  
Kinetic Equation ◽  
Inverse Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Coefficient Inverse Problem ◽  
Space Method

On the basis of a reproducing kernel Hilbert space, reproducing kernel functions for solving the coefficient inverse problem for the kinetic equation are given in this paper. Reproducing kernel functions found in the reproducing kernel Hilbert space imply that they can be considered for solving such inverse problems. We obtain approximate solutions by reproducing kernel functions. We show our results by a table. We prove the eciency of the reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation.

scholarly journals Generalized coherent states, reproducing kernels, and quantum support vector machines

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1292-1306 ◽  
Author(s):  
Rupak Chatterjee ◽  
Ting Yu
Keyword(s):  
Hilbert Space ◽  
Coherent States ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Feature Space ◽  
Reproducing Kernels ◽  
Kernel Functions ◽  
Support Vector ◽  
Machine Learning Classification ◽  
Generalized Coherent States

The support vector machine (SVM) is a popular machine learning classification method which produces a nonlinear decision boundary in a feature space by constructing linear boundaries in a transformed Hilbert space. It is well known that these algorithms when executed on a classical computer do not scale well with the size of the feature space both in terms of data points and dimensionality. One of the most significant limitations of classical algorithms using non-linear kernels is that the kernel function has to be evaluated for all pairs of input feature vectors which themselves may be of substantially high dimension. This can lead to computationally excessive times during training and during the prediction process for a new data point. Here, we propose using both canonical and generalized coherent states to calculate specific nonlinear kernel functions. The key link will be the reproducing kernel Hilbert space (RKHS) property for SVMs that naturally arise from canonical and generalized coherent states. Specifically, we discuss the evaluation of radial kernels through a positive operator valued measure (POVM) on a quantum optical system based on canonical coherent states. A similar procedure may also lead to calculations of kernels not usually used in classical algorithms such as those arising from generalized coherent states.

scholarly journals PIECEWISE OPTIMAL FRACTIONAL REPRODUCING KERNEL SOLUTION AND CONVERGENCE ANALYSIS FOR THE ATANGANA–BALEANU–CAPUTO MODEL OF THE LIENARD’S EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040007
Author(s):  
SHAHER MOMANI ◽  
OMAR ABU ARQUB ◽  
BANAN MAAYAH
Keyword(s):  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Weakly Nonlinear ◽  
Analytical Technique ◽  
Wide Range ◽  
Fractional Models ◽  
Weakly Nonlinear Waves ◽  
Fractional Solution

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on the Fourier approximation theorem to generate a required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of nonfractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.

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